Board Paper of Class 10 2015 Maths (SET 1)  Solutions
General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 31 questions divided into four sections – A, B, C and D.
(iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.
(iv) Use of calculated is not permitted.
(i) All questions are compulsory.
(ii) The question paper consists of 31 questions divided into four sections – A, B, C and D.
(iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.
(iv) Use of calculated is not permitted.
 Question 1
Find the 25^{th} term of the AP $5,\frac{5}{2},0,\frac{5}{2},...$. VIEW SOLUTION
 Question 2
A pole casts a shadow of length $2\sqrt{3}$ m on the ground, when the sun's elevation is 60°. Find the height of the pole. VIEW SOLUTION
 Question 3
A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and these are equally likely outcomes. Find the probability that the arrow will point at any factor of 8. VIEW SOLUTION
 Question 4
Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle. VIEW SOLUTION
 Question 5
In figure 1, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If ∠TPQ = 70°, find ∠TRQ.
VIEW SOLUTION
 Question 6
In Figure 2, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the lengths of TP and TQ.
VIEW SOLUTION
 Question 7
 Question 8
The fourth term of an A.P. is 11. The sum of the fifth and seventh terms of the A.P. is 34. Find its common difference. VIEW SOLUTION
 Question 9
Show that the points (a, a), (–a, –a) and $\left(\sqrt{3}a,\sqrt{3}a\right)$ are the vertices of an equilateral triangle. VIEW SOLUTION
 Question 10
For what values of k are the points (8, 1), (3, –2k) and (k, –5) collinear ? VIEW SOLUTION
 Question 11
Point A lies on the line segment PQ joining P(6, –6) and Q(–4, –1) in such a way that $\frac{\mathrm{PA}}{\mathrm{PQ}}=\frac{2}{5}.$ If point P also lies on the line 3x + k (y + 1) = 0, find the value of k. VIEW SOLUTION
 Question 12
Solve for x :
x^{2} + 5x − (a^{2} + a − 6) = 0 VIEW SOLUTION
 Question 13
In an A.P., if the 12^{th} term is −13 and the sum of its first four terms is 24, find the sum of its first ten terms. VIEW SOLUTION
 Question 14
A bag contains 18 balls out of which x balls are red.
(i) If one ball is drawn at random from the bag, what is the probability that it is not red?
(ii) If 2 more red balls are put in the bag, the probability of drawing a red ball will be $\frac{9}{8}$ times the probability of drawing a red ball in the first case. Find the value of x. VIEW SOLUTION
 Question 15
From the top of a tower of height 50 m, the angles of depression of the top and bottom of a pole are 30° and 45° respectively. Find
(i) how far the pole is from the bottom of a tower,
(ii) the height of the pole. (Use $\sqrt{3}=1.732$) VIEW SOLUTION
 Question 16
The long and short hands of a clock are 6 cm and 4 cm long respectively. Find the sum of the distances travelled by their tips in 24 hours. (Use π = 3.14) VIEW SOLUTION
 Question 17
Two spheres of same metal weigh 1 kg and 7 kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere. VIEW SOLUTION
 Question 18
A metallic cylinder has radius 3 cm and height 5 cm. To reduce its weight, a conical hole is drilled in the cylinder. The conical hole has a radius of $\frac{3}{2}$ cm and its depth is $\frac{8}{9}$ cm. Calculate the ratio of the volume of metal left in the cylinder to the volume of metal taken out in conical shape. VIEW SOLUTION
 Question 19
In Figure 3, ABCD is a trapezium with AB  DC, AB = 18 cm, DC = 32 cm and the distance between AB and DC is 14 cm. If arcs of equal radii 7 cm have been drawn, with centres A,B, C and D, then find the area of the shaded region.
Figure 3 VIEW SOLUTION
 Question 20
A solid rightcircular cone of height 60 cm and radius 30 cm is dropped in a rightcircular cylinder full of water of height 180 cm and radius 60 cm. Find the volume of water left in the cylinder, in cubic metres.
VIEW SOLUTION
$\left[\mathrm{Use}\mathrm{\pi}=\frac{22}{7}\right]$
 Question 21
If x = −2 is a root of the equation 3x^{2} + 7x + p = 0, find the values of k so that the roots of the equation x^{2} + k(4x + k − 1) + p = 0 are equal. VIEW SOLUTION
 Question 22
Find the middle term of the sequence formed by all threedigit numbers which leave a remainder 3, when divided by 4. Also find the sum of all numbers on both sides of the middle term separately. VIEW SOLUTION
 Question 23
The total cost of a certain length of a piece of cloth is Rs 200. If the piece was 5 m longer and each metre of cloth costs Rs 2 less, the cost of the piece would have remained unchanged. How long is the piece and what is its original rate per metre ? VIEW SOLUTION
 Question 24
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. VIEW SOLUTION
 Question 25
In Figure 4, O is the centre of the circle and TP is the tangent to the circle from an external point T. If ∠PBT = 30°, prove that BA : AT = 2 : 1.
Figure 4
 Question 26
Draw a circle of radius 3 cm. From a point P, 7 cm away from its centre draw two tangents to the circle. Measure the length of each tangent. VIEW SOLUTION
 Question 27
Two poles of equal heights are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on the road, the angle of elevation of the top of a pole is 60° and the angle of depression from the top of another pole at point P is 30°. Find the heights of the poles and the distances of the point P from the poles. VIEW SOLUTION
 Question 28
A box contains cards bearing numbers from 6 to 70. If one card is drawn at random from the box, find the probability that it bears
(i) a one digit number.(ii) a number divisible by 5.
(iii) an odd number less than 30.(iv) a composite number between 50 and 70.
VIEW SOLUTION
 Question 29
The base BC of an equilateral triangle ABC lies on yaxis. The coordinates of point C are (0, −3). The origin is the midpoint of the base. Find the coordinates of the points A and B. Also find the coordinates of another point D such that BACD is a rhombus. VIEW SOLUTION
 Question 30
A vessel full of water is in the form of an inverted cone of height 8 cm and the radius of its top, which is open, is 5 cm. 100 spherical lead balls are dropped into the vessel. Onefourth of the water flows out of the vessel. Find the radius of a spherical ball. VIEW SOLUTION
 Question 31
Milk in a container, which is in the form of a frustum of a cone of height 30 cm and the radii of whose lower and upper circular ends are 20 cm and 40 cm respectively, is to be distributed in a camp for flood victims. If this milk is available at the rate of Rs 35 per litre and 880 litres of milk is needed daily for a camp, find how many such containers of milk are needed for a camp and what cost will it put on the donor agency for this. What value is indicated through this by the donor agency ? VIEW SOLUTION
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